Arrow’s Impossibility Theorem

One of the few “useful” theorems that I know about is Arrow’s impossibility theorem. It also highlights a unique feature in mathematics, which is the ability to demonstrating when something is outright impossible. The colloquial manner in which the theorem is described is that no voting system can be “perfect,” where the word “perfect” is

My initial plan today was to read Arrow’s initial paper on the subject “A Difficulty in the Concept of Social Welfare” and give my thoughts, a layman friendly statement, and maybe comments on the proof, but given the fact that I know most of the material, albeit, in an abbreviated form, I did not manage to read through the paper. So, I will just try to provide a layman-friendly (ish) introduction to the statement of the theorem (which you can probably find better explained on YouTube).

The first underlying assumption is that voters have “rational” preferences. This word “rational” is severely (and frustratingly) misunderstood. Here the word means something very specific. Namely that every voter has their own preference ordering for the candidates of an election, which satisfy two conditions:

  1. For each pair of candidates a and b, one of the following is true: a is preferable to b , is preferable to a, or there’s an indifference between a and b.
  2. For any three candidates ab, and c, if a is preferable to b (or there’s indifference between them) and b is not preferable to c (or there’s indifference between them), then is not preferable to c (or there’s indifference between them).

So the first condition states that everyone has a preference or is indifferent between every two possibilities. I’m not a huge fan of the first condition, but I’m willing to go with it since it makes more sense in the context of voting. The second says that your preferences are actually consistent; if you say that you prefer chocolate to strawberry and strawberry to vanilla, then it wouldn’t make sense to go around and claim that you like vanilla more than chocolate.

Now, it becomes a little clearer what we mean by “voting system”; it’s a function that takes as an input the preference orderings of the voters and outputs an ordering for society as a whole. In other words, it’s a process that aggregates everyone’s preferences. Our goal is to make a process that is consistent and reflects, at least a little, the preferences of the voters. So we would want a few criterion, the list that Arrow conceived are as follows:

  1. The voting system has a result for every possibility.
  2. If one candidate x does better (or at least doesn’t do worse) in every voter’s opinion and was previously doing better than y, then x is still doing better than y.
  3. Adding or removing candidates should not affect the relative standing of the rest of the candidates in the total vote.
  4. There are no forced results, or in other words, every candidate has some path to victory.
  5. The election result is not determined by a single person (a dictator).

Arrow’s impossibility theorem states that no voting system can satisfy all five conditions.

Now the voting system that I think is most natural to most people is the plurality (or first-past-the-post) voting system, where everyone votes for one candidate and the one with the most votes wins. The problem with this is, as most people have already guessed, the third condition. If there are two candidates a and b, where a is preferred by more people to b, but then a candidate c, whose policies resemble that of a‘s, enters the race, then the vote is split and b wins the race.

But the problem isn’t just with a single voting system, but with all of them. We have to sacrifice at least one of these conditions whenever we decide on a voting system.


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